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gsw_CT_from_enthalpy.m
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function CT = gsw_CT_from_enthalpy(SA,h,p)
% gsw_CT_from_enthalpy Conservative Temperature from specific
% enthalpy of seawater (75-term equation)
%==========================================================================
%
% USAGE:
% CT = gsw_CT_from_enthalpy(SA,h,p)
%
% DESCRIPTION:
% Calculates the Conservative Temperature of seawater, given the Absolute
% Salinity, specific enthalpy, h, and pressure p. The specific enthalpy
% input is the one calculated from the computationally-efficient
% expression for specific volume in terms of SA, CT and p (Roquet et al.,
% 2015).
%
% Note that the 75-term equation has been fitted in a restricted range of
% parameter space, and is most accurate inside the "oceanographic funnel"
% described in McDougall et al. (2003). The GSW library function
% "gsw_infunnel(SA,CT,p)" is avaialble to be used if one wants to test if
% some of one's data lies outside this "funnel".
%
% INPUT:
% SA = Absolute Salinity [ g/kg ]
% h = specific enthalpy [ J/kg ]
% p = sea pressure [ dbar ]
% ( i.e. absolute pressure - 10.1325 dbar )
%
% SA & h need to have the same dimensions.
% p may have dimensions 1x1 or Mx1 or 1xN or MxN, where SA & h are MxN.
%
% OUTPUT:
% CT = Conservative Temperature ( ITS-90) [ deg C ]
%
% AUTHOR:
% Trevor McDougall and Paul Barker. [ help@teos-10.org ]
%
% VERSION NUMBER: 3.05 (27th January 2015)
%
% REFERENCES:
% IOC, SCOR and IAPSO, 2010: The international thermodynamic equation of
% seawater - 2010: Calculation and use of thermodynamic properties.
% Intergovernmental Oceanographic Commission, Manuals and Guides No. 56,
% UNESCO (English), 196 pp. Available from http://www.TEOS-10.org
%
% McDougall, T.J., 2003: Potential enthalpy: A conservative oceanic
% variable for evaluating heat content and heat fluxes. Journal of
% Physical Oceanography, 33, 945-963.
%
% McDougall, T.J., D.R. Jackett, D.G. Wright and R. Feistel, 2003:
% Accurate and computationally efficient algorithms for potential
% temperature and density of seawater. J. Atmosph. Ocean. Tech., 20,
% pp. 730-741.
%
% McDougall, T.J., and S.J. Wotherspoon, 2014: A simple modification of
% Newton's method to achieve convergence of order 1 + sqrt(2). Applied
% Mathematics Letters, 29, 20-25.
%
% Roquet, F., G. Madec, T.J. McDougall, P.M. Barker, 2015: Accurate
% polynomial expressions for the density and specifc volume of seawater
% using the TEOS-10 standard. Ocean Modelling.
%
% The software is available from http://www.TEOS-10.org
%
%==========================================================================
%--------------------------------------------------------------------------
% Check variables and resize if necessary
%--------------------------------------------------------------------------
if ~(nargin == 3)
error('gsw_CT_from_enthalpy: requires three inputs')
end
[ms,ns] = size(SA);
[mh,nh] = size(h);
[mp,np] = size(p);
if (mh ~= ms | nh ~= ns)
error('gsw_CT_from_enthalpy: SA and h must have same dimensions')
end
if (mp == 1) & (np == 1) % p scalar - fill to size of SA
p = p*ones(ms,ns);
elseif (ns == np) & (mp == 1) % p is row vector,
p = p(ones(1,ms), :); % copy down each column.
elseif (ms == mp) & (np == 1) % p is column vector,
p = p(:,ones(1,ns)); % copy across each row.
elseif (ns == mp) & (np == 1) % p is a transposed row vector,
p = p.'; % transposed then
p = p(ones(1,ms), :); % copy down each column.
elseif (ms == mp) & (ns == np)
% ok
else
error('gsw_CT_from_enthalpy: Inputs array dimensions arguments do not agree')
end %if
if ms == 1
SA = SA.';
h = h.';
p = p.';
transposed = 1;
else
transposed = 0;
end
%--------------------------------------------------------------------------
% Start of the calculation
%--------------------------------------------------------------------------
SA(SA < 0) = 0; % This line ensures that SA is non-negative
CT_freezing = gsw_CT_freezing_poly(SA,p,0); % This is the CT freezing temperature
h_below_freeze = gsw_cp0; % This allows for water to be approx. 1C below the freezing temperature
h_freezing = gsw_enthalpy(SA,CT_freezing,p);
if any(h(:) < (h_freezing(:) - h_below_freeze))
[Icold] = find(h < (h_freezing - h_below_freeze));
h(Icold) = NaN;
% warning('gsw_CT_from_enthalpy: The seawater enthalpy h that was inputed is much less than the enthalpy at the freezing temperature, i.e. the water is frozen, these values have been changed to NaN''s')
end
h_40 = gsw_enthalpy(SA,40*ones(size(SA)),p);
if any(h(:) > h_40(:))
[Ihot] = find(h > h_40);
h(Ihot) = NaN;
% warning('gsw_CT_from_enthalpy: The seawater enthalpy h that was inputed is greater than the enthalpy when CT is 40 C, these values have been changed to NaN''s')
end
CT = CT_freezing + (40 - CT_freezing).*(h - h_freezing)./(h_40 - h_freezing); % First guess of CT
[dummy, h_CT] = gsw_enthalpy_first_derivatives(SA,CT,p);
%--------------------------------------------------------------------------
% Begin the modified Newton-Raphson iterative procedure
%--------------------------------------------------------------------------
CT_old = CT;
f = gsw_enthalpy(SA,CT_old,p) - h;
CT = CT_old - f./h_CT ; % this is half way through the modified Newton's method (McDougall and Wotherspoon, 2014)
CT_mean = 0.5*(CT + CT_old);
[dummy, h_CT] = gsw_enthalpy_first_derivatives(SA,CT_mean,p);
CT = CT_old - f./h_CT ; % this is the end of one full iteration of the modified Newton's method
CT_old = CT;
f = gsw_enthalpy(SA,CT_old,p) - h;
CT = CT_old - f./h_CT ; % this is half way through the modified Newton's method (McDougall and Wotherspoon, 2013)
% After 1.5 iterations of this modified Newton-Raphson iteration,
% the error in CT is no larger than 2.5x10^-14 degrees C, which
% is machine precision for this calculation.
if transposed
CT = CT.';
end
end